The instructions for a lost Re-entry permit or Refugee Travel Document are very simple and very clear. Once you lose your travel document, you must report it to the police in your host country and obtain a police report. Once the police report at hand, then you should go to the US embassy. The embassy will issue you a travel letter as a returning resident. This is an official letter which both airlines and governments worldwide recognize.
It is unlikely that you will be able to board an aircraft with just a greencard. Also remember that the USCIS needs to track down your travel document. To report to the police allows your host country to to keep tract of your document should someone else tries to travel with it. Also to report to your embassy allows the embassy to deactivate or take it out of the system and protects you in the future should your document used for criminal purposes.
As far its effects on citizenship, it has zero if you follow the procedures. But if you don't follow the procedures, nobody knows what might happen